From: hanson@math.uic.edu Date: 20 Nov 2002 20:24:26 -0600 To: nthakr1@uic.edu Subject: Re: computer problem 4 Hello Nishant, You asked: >I need some help with computer problem 4. I did Part A, B and C for 3 given >functions. I don't know how part D works. I tried to look in the book but I >still do not understand it. Can i come see you tommorow for this assignment if >you time. Please email me the time. I free anytime before 12:30 or after 2 pm. >Please let me know. I need some help. Sorry, I only come in MWF and I have office hours right after class at 2pm, but I usually stay any answer questions before I go back to my office. > >About GR3. I quite don't see how is it going to work with the given functions. >Does it have to be between -1 and 1? And if yes, which formula do we use? The problem is that I have not finished talking about Gaussian Quadrature with composite rules in class. The 2XGr_3 means a double application of GR_3 to the interval [a,b], so you have to split that up into two parts, [a,(a+b)/2] and [(a+b)/2,b], split at the midpoint, applying one GR_3 to each part. If we call these part intervals, [a(1),b(1)] and [a(2),b(2)], respectively, that transform x to the universal variable t by the linear transformation given in class, x(t,j) = 0.5*(b(j)-a(j))*t + 0.5*(b(j)+a(j)) for j=1:2 and t in [-1,+1], Then the universal function becomes, F(t,j) = 0.5*(b(j)-a(j))*f(x(t,j)) since dx(t,j)=0.5*(b(j)-a(j)), while the integral of f(x) on [a,b] (there are 3 such functions) becomes, w1*(F(x(t1,1))+F(x(t1,2)))+w2*(F(x(t2,1))+F(x(t2,2)))+w3*(F(x(t3,1))+F(x(t3,2))) where the w's are the Gaussian weights and tj's are the Gaussian nodes given in part D, for j=1:2, including the double application of GR_3. GoodLuck, FBH BCC: Class ============================ Another Version of the Explanation: ============================ From: hanson@math.uic.edu Date: 26 Nov 2002 01:00:37 -0600 To: ssamra1@uic.edu Subject: Re: Problem 4 471 Cc: hanson@math.uic.edu Hello Samir, you said: and it is very confusing? I do not know where you got this from? > In prevouis e-mails, you stated stated the big F is >F(t,j)+0.5*(b(j)-a(j))*f(x(t,j)), this tells me that there are to be two OK, now I see that "+" should be an equals: [a(1),b(1)]=[a,0.5*(a+b)] and [a(2),b(2)]=[0.5*(a+b),b] x(t,j) = 0.5*(b(j)-a(j))*t + 0.5*(b(j)+a(j)) for j=1:2 subintevals. F(t,j) = 0.5*(b(j)-a(j))*f(x(t,j)) for j=1:2 subintevals. >parameters. But later on, when you define the integral for f(x) on [a,b] which >is basically three functions, example w1*(F(x(t1,1)) + F(x(t1,2))), F(x) takes Integral(f,[a,b]) = sum(j=1:2)[Integral(f,[a(j),b(j)]) with approximation: 2XGR3 = sum(j=1:2)[sum(i=1:3)[w_i*F(t_i,j)]], where [w_i]_{3X1} = [w1;w2;w3] and [x_i]_{3X1} = [x1;x2;x3] are the GR3 weights and nodes. Here "i" counts GR3 terms and "j" counts the two subintervals. >only one parameter which is the result of x(t,j). I am confused on what F(x) >takes as a parameter and it's defination. Thank you for your time. GoodLuck, FBH BCC: Class